Harnack inequality for $p$-harmonic functions: improved dimension dependence via tug of war

TL;DR

This paper improves the understanding of Harnack inequalities for p-harmonic functions by using probabilistic tug-of-war methods, resulting in better dimension dependence than traditional approaches.

Contribution

It refines previous tug-of-war analysis to achieve improved dimension dependence in Harnack inequalities for p-harmonic functions.

Findings

Harnack constant is O(exp(C_p d log d)) as dimension d increases

Probabilistic methods yield better constants than Moser iteration

Applicable for all p > 1 in bounded domains

Abstract

Let $p>1$. The Harnack inequality and H\"older continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be used to derive these inequalities. We refine their analysis and obtain improved dependence on $p$ and the dimension $d$ by probabilistic methods. In particular, we show that for all $p>1$, the constant in Harnack's inequality is $O(\exp(C_p d\log d))$ as $d\rightarrow\infty$, which improves the constant derived from Moser iteration.